A384332 Expansion of Product_{k>=1} (1 + k*x)^((2/3)^k).

1, 6, 3, 20, -207, 2538, -36381, 599760, -11210229, 234779146, -5455240455, 139445920452, -3892724842549, 117916363928070, -3854035833235839, 135241405277665656, -5072575747811807052, 202559732310632082120, -8581116791103001216108

Offset: 0

Author

Seiichi Manyama, May 26 2025

Keywords

• sign

Crossrefs

Cf. A116603, A384333, A384334.

Cf. A004123, A384324, A384344.

Programs

Mathematica

terms = 20; A[] = 1; Do[A[x] = -2*A[x] + 3*A[x/(1+x)]^(2/3) * (1+x)^2 + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Vaclav Kotesovec, May 27 2025 *)

PARI

my(N=20, x='x+O('x^N)); Vec(exp(3*sum(k=1, N, (-1)^(k-1)*sum(j=0, k, 2^j*j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = (1+x)^2 * A(x/(1+x))^(2/3).

G.f.: exp(3 * Sum_{k>=1} (-1)^(k-1) * A004123(k+1) * x^k/k).

G.f.: 1/B(-x), where B(x) is the g.f. of A384324.

G.f.: B(x)^6, where B(x) is the g.f. of A384344.

a(n) ~ (-1)^(n+1) * (n-1)! / log(3/2)^(n+1). - Vaclav Kotesovec, May 27 2025